Abstract: In this talk we discuss spectral properties of Schrödinger operators with delta and delta'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the Lipschitz partition and the spectral properties of the corresponding operators are related. As the main result we present an operator inequality for the Schrödinger operators with delta and delta'-interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and it allows to transform known results for Schrödinger operators with delta-interactions to Schrödinger operators with delta'-interactions.
The talk is based on joint work with Pavel Exner and Vladimir Lotoreichik